Reports of the Academy of Sciences of the USSR

1. Volume 139, No. 4

MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR A. V. POGORELOV

ON THE QUESTION OF AN ISOMETRIC IMMERSION OF A TWO-DIMENSIONAL RIEMANNIAN MANIFOLD HOMEOMORPHIC TO A SPHERE INTO A THREE-DIMENSIONAL RIEMANNIAN SPACE

The question of an isometric immersion in the large of a two-dimensional Riemannian manifold homeomorphic to a sphere into a three-dimensional Riemannian space was considered by the author in paper (¹). In that paper a solution of the question was obtained under the assumption that the Gaussian curvature of the immersed manifold is greater than a certain constant depending on the curvature of the space.

In the author’s note (²), this result was improved and, for spaces of nonpositive curvature, assumed in a certain sense a definitive form.

At present the author has succeeded in obtaining a complete solution of the question in the form of the following theorem.

Theorem 1. Let (R) be a complete three-dimensional Riemannian space, and let (M) be a closed Riemannian manifold homeomorphic to a sphere, with Gaussian curvature everywhere greater than a certain constant (c) (greater than, less than, or equal to zero).

Then, if the curvature of the space (R) is everywhere less than (c), (M) admits an isometric immersion into (R) in the form of a regular surface (F).

Moreover, this immersion can be carried out so that a given two-dimensional element (\alpha) of the manifold (M) (a point and a pencil of directions in it) coincides with a prescribed two-dimensional element (\alpha'), isometric to (\alpha), in the space (R), and the surface (F) is situated on the prescribed side of the plane element (\alpha').

If the metric of the space (R) and of the manifold (M) are (k) times differentiable ((k \geqslant 6)), then the surface (F) is differentiable at least (k - 1) times. If the metric of the space (R) and of the manifold (M) are analytic, then the surface (F) is analytic.

The incompleteness of the results obtained in the author’s previous papers (¹,²) is explained mainly by the difficulties arising in obtaining a priori estimates for the normal curvatures of a regular surface in a Riemannian space, depending only on the metric of the surface and the metric of the space. In paper (¹) such estimates are obtained under the assumption that the Gaussian curvature of the surface is greater than a certain constant depending on the curvature of the space, while in paper (²) under the assumption that the Gaussian curvature of the surface is simply greater than the curvature of the space, but the curvature of the space must be everywhere nonpositive. Without going into certain details, for a complete solution of the immersion problem in the form in which we have formulated it, it is sufficient to prove the following proposition concerning a priori estimates.

Let (F) be a locally strictly convex surface in a Riemannian space (R) with regular metric (regularity of the surface itself is not assumed). Let (F_n) be a sequence of regular surfaces converging to (F), and suppose that the metrics of these surfaces converge to the metric of (F) in the class (C^k) ((k \geqslant 6)).

Then, if the Gaussian curvature (F) is strictly greater than the curvature of the space, then for the normal curvatures (F_n), provided (F_n) is sufficiently close to (F), there is an upper estimate independent of (n).

We prove this assertion by contradiction as follows. If the assertion is false, then, without loss of generality, one may assume that on the surface (F_n) there is a point (P_n) at which the normal curvature in one of the directions is greater than (n). Likewise, without loss of generality, one may assume that the points (P_n) converge to a certain point (P_0) of the surface (F).

Take a ball (G_0) with center (P_0) so small that any two of its points are joined by a unique shortest path in (R). Let (\omega_n) be the part of the surface (F_n) contained in the ball (G_0), and let (\omega_0) be the part of (F) inside this ball. Join an arbitrary point (A) of the surface (\omega_0) with the point (P_0). Let (\vartheta(A)) be the angle which the inner normal to (\omega_0) at the point (A) forms with the direction of the shortest path (AP_0). By the strict convexity of (\omega_0), the angle (\vartheta(A)<\pi/2), and for all points (A) that are at distance at least (\varepsilon>0) from (P_0), (\vartheta(A)<\pi/2-\eta(\varepsilon)), where (\eta(\varepsilon)) is positive.

The point (P_0) on the surface (\omega_0) cannot be conical. Therefore there passes through it a certain geodesic (\gamma_0) tangent to the surface (\omega_0). Draw a semigeodesic (\gamma') from the point (P_0) into the body whose boundary is the surface (\omega_0), so that at (P_0) it forms with the geodesic (\gamma_0) an angle less than (\eta(\varepsilon)/2). On the semigeodesic (\gamma') take a point (P'). If the point (P') is sufficiently close to (P_0), and in the definition of the angle (\vartheta(A)) one takes the point (P') instead of (P_0), then this angle, for points (A) removed by a distance greater than (\varepsilon) from (P_0), will also be less than (\pi/2-\eta(\varepsilon)). By virtue of the convergence of (\omega_n) to (\omega_0), for sufficiently large (n) the angle (\vartheta(A)), determined for the surface (\omega_n) and the point (P'), will likewise be less than (\pi/2-\eta(\varepsilon)) when (AP_0<\varepsilon), while for (A\equiv P_n) this angle is greater than (\pi/2-\eta(\varepsilon)/2).

We now define on the surface (\omega_n) the function

[
\overline{w}(X)=\frac{\overline{\varkappa}(X)}{(\cos\vartheta(X))^\mu},
]

where (\overline{\varkappa}(X)) is the maximal normal curvature at the point (X); (\vartheta(X)) is the angle formed by the inner normal to the surface (\omega_n) with the direction of the semigeodesic (XP') at the point (X), and (\mu) is a certain positive constant. If (\omega_n) is sufficiently close to (\omega_0), then for every point (X) satisfying the condition (XP_0>\varepsilon),

[
\overline{w}(X)<\overline{w}(P_n).
]

Indeed, by the definition of the point (P_n), (\overline{\varkappa}(X)\leqslant \overline{\varkappa}(P_n)). Further, (\vartheta(X)<\pi/2-\eta(\varepsilon)), while (\vartheta(P_n)>\pi/2-\eta(\varepsilon)/2). Hence (\vartheta(X)<\vartheta(P_n)), and, consequently, (\overline{w}(X)<\overline{w}(P_n)). Since for (XP_0>\varepsilon) we have (\overline{w}(X)<\overline{w}(P_n)), the maximum of (\overline{w}(X)) is attained at a certain point (X_0) whose distance from the point (P_0) is not greater than (\varepsilon), and therefore whose distance from the point (P') is not greater than (\varepsilon+P_0P').

Now we introduce, in a neighborhood of the point (P'), a polar geodesic coordinate system with pole at (P'). In these coordinates the function (\overline{w}) admits a simple expression. The maximum of (\overline{w}) at the point (X_0) is studied with the aid of the equations of isometric immersion (the analogue of the Darboux equations ((1))). This study, accompanied by rather complicated calculations, makes it possible to establish estimates independent of (n) for the function (\overline{w}) at the point (X_0). At the same time an estimate is obtained for (\overline{\varkappa}(P_n)), and thus we arrive at a contradiction, since by assumption (\overline{\varkappa}(P_n)>n).

Theorem 1 on the possibility of an isometric immersion is supplemented by a theorem on unique determination and isometric transformations.

Theorem 2. The surface (F), whose existence is asserted by Theorem 1, is determined uniquely by the two-dimensional element (a').

Theorem 3. If in a complete Riemannian space (R) with curvature in two-dimensional areas less than (c), two regular isometric surfaces (F_1) and (F_2), homeomorphic to the sphere, are given with Gaussian curvature greater than (c), then one surface admits a continuous bending into the other.

Received
3 V 1961

REFERENCES

1. A. V. Pogorelov, Some Questions of Geometry in the Large in Riemannian Space, Kharkov University Press, 1957.
2. A. V. Pogorelov, DAN, 137, No. 2 (1961).